20-Bio-B4 Robotics · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2015 — 04-Bio-B4 Digital Image Processing. Three hours, open book (any paper notes or textbooks permitted, but no calculator or computer). Six questions of equal value (20 marks each); five constitute a complete paper and only the first five appearing in the answer book are marked. All six are solved here, because this set is a study resource rather than an examination script. Every question is essay/descriptive (definitions, algorithm/system design) except the convolution-size and complexity items in Question 2, the median-filter nonlinearity proof in Question 2(d)(ii), and the illustrative numeric examples worked into Questions 4 and 6.
Reference texts (the books a candidate should have reviewed for this subject):
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
A piezoelectric transducer emits a short high-frequency (typically 2–15 MHz) acoustic pulse into the body and then listens for echoes reflected wherever the acoustic impedance changes (tissue boundaries, organ interfaces). The time delay of each returning echo gives its depth (via the known speed of sound in tissue, ≈1540 m/s), and echo amplitude gives brightness (B-mode imaging); steering/focusing the pulse across an array of elements (or mechanically sweeping a single element) builds up a full 2D cross-sectional image line by line.
T1 (spin-lattice relaxation time) and T2 (spin-spin relaxation time) are the two characteristic time constants governing how excited hydrogen-nucleus spins return to equilibrium after an RF pulse: T1 describes how quickly the net magnetization realigns with the main field (energy given up to the surrounding lattice), while T2 describes how quickly spins lose phase coherence with each other (independent of energy loss to the lattice). Different tissues have characteristically different T1/T2 values, so choosing a pulse sequence/timing that is weighted toward T1 or T2 produces very different image contrast between the same tissues (e.g. fat is bright on T1-weighted images, fluid/CSF is bright on T2-weighted images).
The Radon transform maps a 2D function $f(x,y)$ to the set of its line integrals along every possible line, parameterized by angle $\theta$ and perpendicular offset $t$:
$$p(t,\theta)=\int\!\!\int f(x,y)\,\delta(x\cos\theta+y\sin\theta-t)\,dx\,dy$$It is significant to medical imaging because it is exactly the physical measurement an X-ray CT scanner makes: each detector reading is the line integral of tissue attenuation along the ray path, and a full CT acquisition collects $p(t,\theta)$ for many angles $\theta$ (the resulting 2D array of projections is the sinogram). Reconstructing the cross-sectional image $f(x,y)$ from the measured projections is therefore an inverse Radon transform problem (solved in practice by filtered back-projection or iterative reconstruction), making the Radon transform the mathematical foundation of tomographic image formation.
| X-ray CT | MRI | |
|---|---|---|
| i. For the patient | Fast (seconds); uses ionizing radiation (cumulative cancer risk, avoided in pregnancy); generally cheaper and more widely available; contraindicated less often | No ionizing radiation; much slower (many minutes), requiring the patient to stay still; contraindicated with certain implants/pacemakers (strong magnetic field); can be loud and induce claustrophobia |
| ii. Acquired images | Excellent for dense/bony structures and acute haemorrhage; attenuation contrast is poor between many soft tissues; single physical contrast mechanism (attenuation coefficient) | Excellent soft-tissue contrast (multiple contrast mechanisms: T1, T2, proton density, diffusion, functional); poor imaging of cortical bone/lung air spaces (low proton density) |
Speckle is a granular, multiplicative interference pattern that arises whenever coherent radiation (ultrasound, laser, synthetic-aperture radar) is reflected from a surface or medium rough at the scale of the wavelength: many sub-resolution scatterers within one resolution cell return waves that interfere constructively or destructively depending on their random relative phases, producing a signal-dependent granular texture rather than additive noise. It is characteristic of ultrasound B-mode images, laser speckle, and SAR imagery — and, being multiplicative, is best modelled as $g=f\cdot n$ (Question 4(a)), not the additive model appropriate for incoherent-imaging noise.
MRI spatial resolution is limited by: the strength and linearity of the applied magnetic-field gradients (steeper, more linear gradients allow finer spatial encoding); the signal-to-noise ratio available (finer voxels contain fewer excited spins, so shrinking voxel size trades away SNR, ultimately bounded by the main field strength and coil sensitivity); acquisition time constraints (finer k-space sampling for higher resolution takes longer, competing against patient motion and practical scan-time limits); and patient/physiological motion (cardiac, respiratory, and bulk motion blur fine detail unless specifically gated or corrected for).
Extending from 2D to 3D multiplies the data volume by the number of slices, so storage, memory, and processing time for filtering/segmentation/visualization all grow substantially (and naïve extensions of 2D algorithms, e.g. 3D connected components or 3D convolution, cost far more per voxel than their 2D analogues). Acquisition itself is harder: 3D volumes take proportionally longer to scan, increasing exposure to motion artifacts across the whole volume rather than a single slice, and true 3D reconstruction (as opposed to a stack of independently acquired 2D slices) must contend with anisotropic resolution (slice thickness typically far coarser than in-plane resolution) and partial-volume effects at slice boundaries. Finally, visualization and interpretation are themselves harder in 3D — rendering/segmenting a volumetric structure and displaying it meaningfully to a clinician (surface rendering, multi-planar reformatting) is a substantially harder problem than displaying a single 2D slice.
